L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.407 + 1.68i)3-s + 1.00i·4-s + (0.707 − 0.707i)5-s + (1.47 − 0.902i)6-s + 2.94·7-s + (0.707 − 0.707i)8-s + (−2.66 − 1.37i)9-s − 1.00·10-s − 5.04·11-s + (−1.68 − 0.407i)12-s + (2.73 + 2.73i)13-s + (−2.07 − 2.07i)14-s + (0.902 + 1.47i)15-s − 1.00·16-s + (−5.09 + 5.09i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (−0.235 + 0.971i)3-s + 0.500i·4-s + (0.316 − 0.316i)5-s + (0.603 − 0.368i)6-s + 1.11·7-s + (0.250 − 0.250i)8-s + (−0.889 − 0.457i)9-s − 0.316·10-s − 1.52·11-s + (−0.485 − 0.117i)12-s + (0.758 + 0.758i)13-s + (−0.555 − 0.555i)14-s + (0.232 + 0.381i)15-s − 0.250·16-s + (−1.23 + 1.23i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.141 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.141 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9570060238\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9570060238\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (0.407 - 1.68i)T \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
| 37 | \( 1 + (5.76 + 1.93i)T \) |
good | 7 | \( 1 - 2.94T + 7T^{2} \) |
| 11 | \( 1 + 5.04T + 11T^{2} \) |
| 13 | \( 1 + (-2.73 - 2.73i)T + 13iT^{2} \) |
| 17 | \( 1 + (5.09 - 5.09i)T - 17iT^{2} \) |
| 19 | \( 1 + (-2.15 - 2.15i)T + 19iT^{2} \) |
| 23 | \( 1 + (-3.53 + 3.53i)T - 23iT^{2} \) |
| 29 | \( 1 + (-4.90 - 4.90i)T + 29iT^{2} \) |
| 31 | \( 1 + (4.78 - 4.78i)T - 31iT^{2} \) |
| 41 | \( 1 - 5.08T + 41T^{2} \) |
| 43 | \( 1 + (1.16 + 1.16i)T + 43iT^{2} \) |
| 47 | \( 1 - 6.36iT - 47T^{2} \) |
| 53 | \( 1 - 3.67iT - 53T^{2} \) |
| 59 | \( 1 + (6.34 - 6.34i)T - 59iT^{2} \) |
| 61 | \( 1 + (-3.36 + 3.36i)T - 61iT^{2} \) |
| 67 | \( 1 - 11.8iT - 67T^{2} \) |
| 71 | \( 1 - 6.94iT - 71T^{2} \) |
| 73 | \( 1 + 8.41iT - 73T^{2} \) |
| 79 | \( 1 + (-10.3 - 10.3i)T + 79iT^{2} \) |
| 83 | \( 1 - 4.78iT - 83T^{2} \) |
| 89 | \( 1 + (2.54 + 2.54i)T + 89iT^{2} \) |
| 97 | \( 1 + (9.97 + 9.97i)T + 97iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.36737096631391991166154335193, −9.137824048446037760660983769792, −8.645530968693853814021987035323, −8.030707250146900146369872592150, −6.72282531953146038166986067367, −5.54711827640432961392894597640, −4.80847850142146220148255871415, −4.00157368500669505420119367769, −2.71322388196575191870196163370, −1.49930670795466634652165909409,
0.51793879155400736481907719605, 1.95625283149115142992809060645, 2.88860861296958081285488076826, 4.96344015230763602626636402226, 5.36657999875725484395840992267, 6.40239461560434064809518855402, 7.31457649263899105679753987182, 7.84094011275792086229413518510, 8.516468568719562115902174118635, 9.456226615774883134452186961570